Author(s): Oscar Zariski, Pierre Samuel
Series: University Series in Higher Mathematics
Edition: 6th printing, Jan 1967
Publisher: D. Van Nostrand Company
Year: 1967
Language: English
Commentary: new scan
Pages: 329+xi
City: Toronto, London
Title
Preface
Contents
I. Introductory concepts
§ 1. Binary operations
§ 2. Groups
§ 3. Subgroups
§ 4. Abelian groups
§ 5. Rings
§ 6. Rings with identity
§ 7. Powers and multiples
§ 8. Fields
§ 9. Subrings and subfields
§ 10. Transformations and mappings
§ 11. Group homomorphisms
§ 12. Ring homomorphisms
§ 13. Identification of rings
§ 14. Unique factorization domains
§ 15. Euclidean domains
§ 16. Polynomials in one indeterminate
§ 17. Polynomial rings
§ 18. Polynomials in several indeterminates
§ 19. Quotient fields and total quotient rings
§ 20. Quotient rings with respect to multiplicative systems
§ 21. Vector spaces
II. Elements of field theory
§ 1. Field extensions
§ 2. Algebraic quantities
§ 3. Algebraic extensions
§ 4. The characteristic of a field
§ 5. Separable and inseparable algebraic extension
§ 6. Splitting fields and normal extensions
§ 7. The fundamental theorem of Galois theory
§ 8. Galois fields
§ 9. The theorem of the primitive element
§ 10. Field polynomials. Norms and traces
§ 11. The discriminant
§ 12. Transcendental extensions
§ 13. Separably generated fields of algebraic functions
§ 14. Algebraically closed fields
§ 15. Linear disjointness and separability
§ 16. Order of inseparability of a field of algebraic functions
§ 17. Derivations
III. Ideals and modules
§ 1. Ideals and modules
§ 2. Operations on submodules
§ 3. Operator homomorphisms and difference modules
§ 4. The isomorphism theorems
§ 5. Ring homomorphisms and residue class rings
§ 6. The order of a subset of a module
§ 7. Operations on ideals
§ 8. Prime and maximal ideals
§ 9. Primary ideals
§ 10. Finiteness conditions
§ 11. Composition series
§ 12. Direct sums
§ 12bis. Infinite direct sums
§ 13. Comaximal ideals and direct sums of ideals
§ 14. Tensor products of rings
§ 15. Free joins of integral domains (or of fields)
IV. Noetherian rings
§ 1. Definitions. The Hilbert basis theorem
§ 2. Rings with descending chain condition
§ 3. Primary rings
§ 3bis. Alternative method for studying the rings with d.c.c.
§ 4. The Lasker-Noether decomposition theorem
§ 5. Uniqueness theorems
§ 6. Application to zero-divisors and nilpotent elements
§ 7. Application to the intersection of the powers of an ideal
§ 8. Extended and contracted ideals
§ 9. Quotient rings
§ 10. Relations between ideals in R and ideals in R_M
§ 11. Examples and applications of quotient rings
§ 12. Symbolic powers
§ 13. Length of an ideal
§ 14. Prime ideals in noetherian rings
§ 15. Principal ideal rings
§ 16. Irreducible ideals
Appendix: Primary representation in Noetherian modules
V. Dedekind domains, classical ideal theory
§ 1. Integral elements
§ 2. Integrally dependent rings
§ 3. Integrally closed rings
§ 4. Finiteness theorems
§ 5. The conductor of an integral closure
§ 6. Characterizations of Dedekind domains
§ 7. Further properties of Dedekind domains
§ 8. Extensions of Dedekind domains
§ 9. Decomposition of prime ideals in extensions of Dedekind domains
§ 10. Decomposition group, inertia group, and ramification groups
§ 11. Different and discriminant
§ 12. Application to quadratic fields and cyclotomic fields
§ 13. A theorem of Kummer
Index of notations
Index of definitions
